Calculus- Early Transcendentals, 2021a

6 Review
1.1.3 The Quadratic Formula and Completing the Square
The technique of completing the square allows us to solve quadratic equations and also to determine the
center of a circle/ellipse or the vertex of a parabola.
The main idea behind completing the square is to turn:
ax 2 + bx + c
into
a(x − h) 2 + k
One way to complete the square is to use the following formula:
(
ax 2 + bx + c = a x + b ) 2
− b2
2a 4a 2 + c
But this formula is a bit complicated, so some students prefer following the steps outlined in the next
example.
Example 1.8: Completing the Square
Solve 2x 2 + 12x − 32 = 0 by completing the square.
Solution. In this instance, we will not divide by 2 first (usually you would) in order to demonstrate what
you should do when the ‘a’ value is not 1.
2x 2 + 12x − 32 = 0
2x 2 + 12x = 32
Start with original equation.
Move the number over to the other side.
2(x 2 + 6x)=32 Factor out the a from the ax 2 + bx expression.
6 → 6 2 = 3 → 32 = 9 Take the number in front of x,
divide by 2,
then square it.
2(x 2 + 6x + 9)=32 + 2 · 9 Add the result to both sides,
taking a = 2 into account.
2(x + 3) 2 = 50 Factor the resulting perfect square trinomial.
You have now completed the square!
(x + 3) 2 = 25 → x = 2orx = −8 To solveforx, simply divide by a = 2
and take square roots.